Properties

Label 980.403
Modulus $980$
Conductor $980$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,63,80]))
 
pari: [g,chi] = znchar(Mod(403,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 980.bs

\(\chi_{980}(23,\cdot)\) \(\chi_{980}(107,\cdot)\) \(\chi_{980}(123,\cdot)\) \(\chi_{980}(163,\cdot)\) \(\chi_{980}(207,\cdot)\) \(\chi_{980}(247,\cdot)\) \(\chi_{980}(303,\cdot)\) \(\chi_{980}(347,\cdot)\) \(\chi_{980}(387,\cdot)\) \(\chi_{980}(403,\cdot)\) \(\chi_{980}(443,\cdot)\) \(\chi_{980}(487,\cdot)\) \(\chi_{980}(527,\cdot)\) \(\chi_{980}(543,\cdot)\) \(\chi_{980}(583,\cdot)\) \(\chi_{980}(627,\cdot)\) \(\chi_{980}(683,\cdot)\) \(\chi_{980}(723,\cdot)\) \(\chi_{980}(767,\cdot)\) \(\chi_{980}(807,\cdot)\) \(\chi_{980}(823,\cdot)\) \(\chi_{980}(907,\cdot)\) \(\chi_{980}(947,\cdot)\) \(\chi_{980}(963,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((491,197,101)\) → \((-1,-i,e\left(\frac{20}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(403, a) \) \(1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 980 }(403,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(403,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(403,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(403,·)) \;\) at \(\; a,b = \) e.g. 1,2